Number 22 linear algebra V and if T is a set of more than p vectors in V,mensional vectorthen T is linearly dependent.20. a. R2 is a two-dimensional subspace of R'.b. The number of variables in the equation Ax =0 equalsthe dimension of Nul A.= = %}c. A vector space is infinite-dimensional if it is spanned byan infinite set.d. If dim V =n and if S spans V, then S is a basis of V.s in R' whosee. The only three-dimensional subspace of R' is R' itself.21. The first four Hermite polynomials are 1, 2t, -2+ 4t2, and-12t + 813. These polynomials arise naturally in the studyof certain important differential equations in mathematicalphysics.? Show that the first four Hermite polynomials forma basis of P3.2spanned bythe subspace22. The first four Laguerre polynomials are 1, 1-1,2-4t +1,and 6 18t +9t2-t. Show that these polynomials form abasis of P3.23. Let B be the basis of P, consisting of the Hermite polynomi-als in Exercise 21, and let p(t) = -1+8t? + 8t³. Find thecoordinate vector of p relative to B.24. Let B be the basis of P, consisting of the first threeLaguerre polynomials listed in Exercise 22, and letp(t) = 5+5t - 2t2. Find the coordinate vector of p relativeFor the matricesto B.25. Let S be a subset of an n-dimensional vector space V, andsuppose S contains fewer than n vectors. Explain why Scannot span V.26. Let H be an n-dimensional subspace of an n-dimensionalvector space V. Show that H =V.27. Explain why the space P of all polynomials is an infinite-dimensional space.2 See Introduction to Functional Analysis, 2d ed., by A. E, Taylor andDavid C. Lay (New York: John Wiley & Sons, 1980), pp. 92-93. Othersets of polynomials are discussed there, too.3.

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Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter5: Inner Product Spaces

Section5.CR: Review Exercises

Problem 42CR: Repeat Exercise 41 for B={(1,2,2),(1,0,0)} and x=(3,4,4). Let B={(0,2,2),(1,0,2)} be a basis for a...

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Number 22 linear algebra

V and if T is a set of more than p vectors in V,
mensional vector
then T is linearly dependent.
20. a. R2 is a two-dimensional subspace of R'.
b. The number of variables in the equation Ax =0 equals
the dimension of Nul A.
= = %}
c. A vector space is infinite-dimensional if it is spanned by
an infinite set.
d. If dim V =n and if S spans V, then S is a basis of V.
s in R' whose
e. The only three-dimensional subspace of R' is R' itself.
21. The first four Hermite polynomials are 1, 2t, -2+ 4t2, and
-12t + 813. These polynomials arise naturally in the study
of certain important differential equations in mathematical
physics.? Show that the first four Hermite polynomials form
a basis of P3.
2
spanned by
the subspace
22. The first four Laguerre polynomials are 1, 1-1,2-4t +1,
and 6 18t +9t2-t. Show that these polynomials form a
basis of P3.
23. Let B be the basis of P, consisting of the Hermite polynomi-
als in Exercise 21, and let p(t) = -1+8t? + 8t³. Find the
coordinate vector of p relative to B.
24. Let B be the basis of P, consisting of the first three
Laguerre polynomials listed in Exercise 22, and let
p(t) = 5+5t - 2t2. Find the coordinate vector of p relative
For the matrices
to B.
25. Let S be a subset of an n-dimensional vector space V, and
suppose S contains fewer than n vectors. Explain why S
cannot span V.
26. Let H be an n-dimensional subspace of an n-dimensional
vector space V. Show that H =V.
27. Explain why the space P of all polynomials is an infinite-
dimensional space.
2 See Introduction to Functional Analysis, 2d ed., by A. E, Taylor and
David C. Lay (New York: John Wiley & Sons, 1980), pp. 92-93. Other
sets of polynomials are discussed there, too.
3.

Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.

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